Piotr Liczberski's research while affiliated with Lodz University of Technology and other places

Publications (51)

Article
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In the paper there is considered a generalization of the well-known Fekete–Szegö type problem onto some Bavrin’s families of complex valued holomorphic functions of several variables. The definitions of Bavrin’s families correspond to geometric properties of univalent functions of a complex variable, like as starlikeness and convexity. First of all...
Article
Full-text available
This paper is devoted to a generalization of the well-known Fekete-Szegö type coefficients problem for holomorphic functions of a complex variable onto holomorphic functions of several variables. The considerations concern three families of such functions f, which are bounded, having positive real part and which Temljakov transform Lf has positive...
Article
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The paper concerns holomorphic functions in complete bounded n-circular domains \({{\mathcal {G}}}\) of the space \({\mathbb {C}}^n\). The object of the present investigation is to solve majorization problem of Temljakov operator. This type of problem has been studied earlier in Liczberski and Żywień (Folia Sci Univ Tech Res 33:37–42, 1986), Liczbe...
Article
The paper is devoted to the investigations of holomorphic functions on complete n-circular domains G of ℂn which are solutions of some partial differential equations in G. Our considerations concern a collection M Gk, k ≥ 2, of holomorphic solutions of equations corresponding to planar Sakaguchi’s conditions for starlikeness with respect to k-symme...
Article
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The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families \(\mathcal {C}_{\mathcal {G}},\mathcal {M}_{\mathcal {G}},\mathcal {N}_{\mathcal {G}},\mathcal {R}_{\mathcal {G}},\mathcal {V}_{\mathcal {G}}\)...
Article
The subject of the paper concerns the existence of different inclusions between families St, Sc of C-n - biholomorphic starlike or convex mappings and a family S(k), k >= 2, of C-n - locally biholomorphic mappings. In the definition of S(k) we used a combination of a Kikuchi-Matsuno-Snifridge's starlikeness condition and a property of (j, k)-symmet...
Article
In this paper some characterizations and properties of domains in Re with boundary accessible externally by some circular cones are given. It is shown that the boundary of such domains is also conically accessible from the interior. It is also characterized alpha-accessible domains of C-N, which are biholomorphically equivalent to the open Euclidea...
Article
We continue a concept of strongly starlike mappings of order alpha on bounded balanced pseudoconvex domains with C 1 -plurisubharmonic defining function, introduced by H. Hamada, G. Kohr and M. Kohr [Rev. Roum. Math. Pures Appl. 44, No. 4, 583–594 (1999; Zbl 0999.32008)]. The main theorem contains two new geometric properties of such mappings and g...
Article
In the paper an internal geometric characterization of strongly starlike mappings of order alpha in CnCn is given.
Article
In the paper necessary and sufficient conditions for the boundary starlikeness of holomorphic mappings in Cn are given. In the proof an n-dimensional version of Julia's theorem is used.
Article
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The paper concerns holomorphic functions in complete bounded n-circular domains of the space ℂⁿ and presents some properties of the above mentioned functions belonging to the families described by some geometrical or analytical conditions. This subject has been considered by many mathematicans, for example I.I. Bavrin, K. Dobrowolska, I. Dziubiński...
Article
In this paper we offer new results of research presented in the referred papers of J.E. Fornaess and E.L. Stout, E. Ligocka and the authors, and concerning the existence of m-valent locally biholomorphic mappings from product domains of Cn onto n-dimensional complex manifolds. In particular, we confirm an own conjecture about the estimation of the...
Article
We continue E. Ligocka’s [Ann. Pol. Math. 82, No. 2, 127–135 (2003; Zbl 1057.30019)] investigations concerning the existence of m-valent locally biholomorphic mappings from multi-connected onto simply connected domains. We decrease the constant m, and also give the minimum of m in the case of mappings from a wide class of domains onto the complex p...
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In this paper, we consider the problem of distortion theorems for mappings which map the unit ball biholomorphically onto convex domains in ℂ n . In particular, we discuss two distortion conjectures for such mappings.
Article
In the paper there are given new applications of dimen- sion reduction method (from (3)) of studing linearly invariant families of holomorphic mappings in n. There also included generalizing mod- ications of this method.
Article
In the paper the decomposition with respect to the group of roots of unity is applied to characterize some classes of biholomorphic mappings in There is considered the problem of subordination and majorization relations between convex mappings and their components in the above partition. There is also given a distortion theorem for convex mappings...
Article
In the paper the problem of sharp lower estimation for ‖Df(z)‖ in the class of normalized biholomorphic mappings f between the open unit ball Bn and convex domains in Cn has been considered.
Article
In the present paper we give a method of obtaining some solutions for functional equations in which the unknown function occurs in the form of its own iterates. We reduce the equation to one of the type we have solved in [2] using properties of (j, k)-symmetrical functions, which are collected in [1].
Article
We continue our previous work on a problem of Janiec connected with a uniqueness theorem, of Cartan{Gutzmer type, for holomorphic mappings in C n . To solve this problem we apply properties of (j;k)-symmetrical functions.
Article
The notion of a linearly invariant family of mappings of a ball in C was introduced in the article ``Pfaltzgraff J. A., Distortion of locally biholomorphic maps of the n-ball, Complex Variables, 33, 239-253 (1997)''''; it generalizes the classical case n=1 studied earlier by Ch. Pommerenke and other authors. In the indicated article, Pfaltzgraff in...
Article
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In this paper, we will give the growth theorem of starlike mappings of order on the unit ball B in complex Banach spaces. We also give an analytic sucien t condition for a locally biholomorphic mapping on B to be a starlike mapping of order .
Article
In this paper we suggest a new definition of the order of a linearly invariant family of locally biholomorphic mappings of the unit ball in . This definition is equivalent to the one given by Pfaltzgraff in [J.A. PfaltzgraffComplex Variables Theory Appl 33 (1997), 239–253.]. It bases on a very simple relationship with the Jacobian of the mappings (...
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In this paper, we will consider classes of subordinations involving partial derivatives of holomorphic mappings in complex Banach spaces.
Article
In this paper, we will consider some sufficient conditions for locally biholomorphic mappings defined in the unit ball in complex Banach spaces to be biholomorphic and to have Φ-like images. As a corollary, we obtain some sufficient condition for locally biholomorphic mappings to be starlike mappings.
Article
In a recent paper, Gurganus extended Brickman's work on Φ-like holomorphic functions on the unit disc in C to locally biholomorphic mappings on the unit ball in a Banach space. In this paper, we will extend the above results to locally biholomorphic mappings on bounded balanced pseudoconvex domains with C 1 plurisubharmonic defining functions in C...
Article
In this paper we introduce the concept of strongly starlikeness of order α > 0, for holomorphic mappings defined on the unit ball of Cn. We obtain the distorsion and the covering theorems for strongly starlike mappings of order α ∈ (0,1] and we give a connection between strongly starlikeness and spirallikeness in Cn. http://web.math.hr/glasnik/vol_...
Article
Introduction. In the present paper the authors study some families of functions from a complex linear space X into a complex linear space Y . They introduce the notion of (j, k)-symmetrical function (k = 2, 3, . . .; j = 0, 1, . . . , k - 1) which is a generalization of the notions of even, odd and k-symmetrical functions. It has turned out that fo...
Article
P. T. Mocanu in the paper [3] has shown that the well known univalence criterions for holomorphic functions of one variable, due to W. Alexander W. Kaplan, K. Noshiro, S. Ozaki, S. E. Warschawski and J. Wolf (see [1], [2], [4], [5], [8], [9]) can be extended onto the complex functions from the class C . The same way, in n-dimensional case, we gener...
Article
We give a version of the distortion theorem for biholomorphic mappings between the unit polydisc and convex domains in the space of n complex variables.
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The author gives a simple starlikeness criterion for locally biholomorphic mapping from the open unit ball in C n into C n.
Article
In the recent paper [ibid. 103, Mat. Fiz. 16,. Mat. 12, 103-113 (1992)] we introduced a generalization of the notions of even functions and odd functions, proved several properties of these notions and gave its different applications. In the present paper we present several new applications of these notions for the investigation of the solutions of...
Article
In this paper we introduce the concept of strongly starlikeness of order Ct. > 0, for holomorphic mappings defined on the unit ball of en, We obtain the distortion and the covering theorems for strongly starlike mappings of order Ct. E (0, 1) and we give a connection between strongly starlikeness and spirallikeness in en, j=1 Z E en. The open Eucli...
Article
Full-text available
In this paper, we will give the growth theorem of starlike mappings of order α on the unit ball B in complex Banach spaces. We also give an analytic sufficient condition for a locally biholomorphic mapping on B to be a starlike mapping of order α. http://web.math.hr/glasnik/vol_36/no1_05.html

Citations

... Now, we show that in the case | | < 1, the equality in (2.1) realizes the function =̂ ∈ B G (0),̂ = I 2k , more precisely ̂ = I 2k |G , i.e., ̂ (z) = I 2k (z), z ∈ G. Indeed, since Q̂ ,k = 0, Q̂ ,2k = I 2k , we get This completes the proof. ◻ Note that Theorem 2.1 generalizes an earlier result given by authors in [7]. ...
... F k represents the family including all k-fold symmetric functions. e concept of k-symmetrical function was protracted to so-called (j, k)-symmetrical function by Liczberski and Połubiński in [2]. To be specific, a function f(ξ) is reported for being (j, k)-symmetrical if ...
... More precisely, the function f of the right hand side in (2.5) is replaced by a function from F 0,k (G,ℂ), generated by f. Formally, we say that a function f ∈ H G (1) belongs to M k G , k ∈ ℕ 2 , (see [3,5]) if there exists a function h ∈ C G such that where f 0,k ∈ F 0,k (G,ℂ) is the (0, k) -symmetrical component of the function f in the decomposition (1.2) from Theorem A. This family for k = 2 corresponds to a well-known Sakaguchi family [27] of a complex variable functions; strictly speaking of functions univalent starlike with respect to two symmetric points. In the paper [5] it was shown that for k ∈ ℕ ...
... We say that a function f ∈ H G (1) belongs to the Bavrin's family M G (N G ) if it satisfies the factorization together with a function h ∈ C G and the transform L f , (L L f ), respectively. Note that the families M G , N G correspond with the well-known families of normalized univalent starlike (convex) functions in the disc U [2] and the family M G can be used to construction biholomorphic starlike mappings in ℂ n (see [6,18], compare also [12,25]). Between functions ...
... In [26] the notion of α-accessible domain was defined. It was shown that such domains posses the cone condition with l(p) = −p. ...
... First, we will generalize Lemma 2.3 to complex Banach spaces (cf. [14], [15]). The following theorem generalizes the result of third author's paper [16]. ...
... Condition (1) is equivalent to » ( ! ) ( 2 ) R e f f l £ 4 > t a n obviously equivalent to (!) Im /'(* )(« ) for every z £ D and a? ^ 0, which is For <p = G we obtain Theorem 1 and Theorem 2 from [1]; for other extensions in the case <p = 0 see [2], [3], [4] and [5]. As in [1], we use the following theorem Using Theorem 2, we obtain that f n is injective on D for n G N and since f n -► / uniformly on the compact subsets of D , we see that / is injective on D . ...
... In [6] (see also [7]) α-accessible domain, α ∈ [0, 1), were introduced and studied. A domain Ω ⊂ R n , 0 ∈ Ω, is called αaccessible, if for every point p ∈ ∂Ω there exists a number r = r(p) > 0 such that the cone ...
... In the next sections of the paper will be very useful the fact that f 0,k ∈ H G (1) for f ∈ H G (1). Note that Note that the above unique decomposition (1.2) of functions was used in [20] to solve some functional equations, in [21] to construction a semi power series and in [22] to obtain a uniqueness theorem of Cartan type for holomorphic mappings in ℂ n . ...
... then ord f r ≤ α (see [5]). Now Theorem 2 will be applied to obtaining a simple direct proof of the following result connected with universal linearly invariant families U n α . ...